Explain precision/recall and compute NN output

You are given a short ML fundamentals assessment with three parts. ## Part A — Precision/Recall/F1 A binary classifier on a dataset produced the following confusion-matrix counts: - True Positives (TP) = 40 - False Positives (FP) = 10 - False Negatives (FN) = 20 - True Negatives (TN) = 130 1. Compute **precision**, **recall**, and **F1**. 2. If you **raise the decision threshold**, what typically happens to precision and recall (and why)? 3. In an **imbalanced** dataset where positives are rare, when would you optimize for precision vs. recall? ## Part B — Ensemble learning (select all correct) For each statement, mark whether it is generally **True** or **False**, and briefly justify. 1. Bagging primarily reduces **variance**. 2. Bagging uses **bootstrap sampling** (sampling with replacement) to train each base model. 3. Boosting trains base learners **independently** and can be fully parallelized without changing the algorithm. 4. Ensembles can improve generalization because averaging/voting can cancel out uncorrelated errors. 5. If the base learners are perfectly correlated (always make the same predictions), bagging will provide large gains. ## Part C — Forward pass of a small neural network Compute the network output for the following fully connected network. Use the sigmoid activation \(\sigma(t)=\frac{1}{1+e^{-t}}\) at **both** layers. - Input \(x = [1.0,\ 2.0]^T\) - Hidden layer (2 units): \(h = \sigma(W_1 x + b_1)\) - \(W_1 = \begin{bmatrix}0.5 & -1.0\\ 1.0 & 0.5\end{bmatrix}\), \(b_1 = \begin{bmatrix}0.0\\ -0.5\end{bmatrix}\) - Output layer (1 unit): \(\hat{y} = \sigma(W_2 h + b_2)\) - \(W_2 = \begin{bmatrix}1.5 & -2.0\end{bmatrix}\), \(b_2 = 0.1\) Return \(\hat{y}\) as a decimal rounded to 4 digits.